Try opening Keychain.app and removing all certificates that could possibly be related to this. Also, open Utilities » Java Preferences and show us the output from the Security tab. You could also clear the Java cache and files from the Network tab here. Furthermore, which version of OS X are you using? Which page are you trying to access exactly (so we can try to reproduce the issue)? Which Java version are you running?
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slhck♦Jul 31 '12 at 22:35

I've just installed OS X Mountain Lion today, and I got at least the downloadable minecraft to run (it didn't before). Hooray! Since it completely uninstalled Java, it installed the latest version (Java SE 6). I'll give more info if none of the existing answers do the trick.
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JwostyJul 31 '12 at 22:49

2 Answers
2

Go into System preferences -> Security and Privacy and check the box that allows applications to be downloaded from anywhere (click the lock to make changes to this).

If it doesn't help, you can try as follows:
Make sure that you don't any other windows apart this certificate, maybe it's asking for you permission in the separate window.
Try as well clearing your java cache and security settings.

You may also need to reinstall the locally installed Minecraft files which you can find in your home directory.

On Windows: %AppData% (Windows+R keys, type in %AppData% at the run command and press Enter).

On Mac: ~/Library/Application Support/minecraft (your local home user directory).

There will be a minecraft directory, so try to delete everything except for the saves directory that you see therein. Then, try running Minecraft again. I'd advise you to make the backup of those files before. If you don't have any Minecraft data stored, then remove the whole minecraft directory.

Troubleshooting:

make sure that your system date is not in the past,

run your Chrome in the terminal, it could give you some more information about the problem,